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In this article, we study the structure of a multiple variable mapping. Indeed, we reduce the system of several mixed additive-quadratic equations defining a multivariable mapping to obtain a single functional equation, say, the multimixed additive-quadratic equation. We also show that such mappings under some conditions can be multi-additive, multi-quadratic and multi-additive-quadratic. Moreover, we establish the Hyers–Ulam stability of the multimixed additive-quadratic equation, using the so-called direct (Hyers) method. Additionally, we present a concrete example (the numerical approximation) regarding the stability of some two variable mappings into real numbers. Applying some characterization results, we indicate two examples for the case that a multimixed additive-quadratic mapping (in the special cases) cannot be stable. [ABSTRACT FROM AUTHOR]

In this article, with simple and short proofs without applying fixed point theorems, some hyperstability results corresponding to the functional equations of Cauchy and Jensen are presented in 2-normed spaces. We also obtain some results on hyperstability for the general linear functional equation f (a x + b y) = A f (x) + B f (y) + C . [ABSTRACT FROM AUTHOR]

In this paper, we solve the system of functional equations { f (x + y) + g (y − x) = 2 f (x) , g (x + y) − f (y − x) = 2 g (y) and we investigate the stability of g-derivations in Banach algebras. [ABSTRACT FROM AUTHOR]

To make better decisions on approximation, we may need to increase reliable and useful information on different aspects of approximation. To enhance information about the quality and certainty of approximating the solution of an Apollonius-type quadratic functional equation, we need to measure both the quality and the certainty of the approximation and the maximum errors. To measure the quality of it, we use fuzzy sets, and to achieve its certainty, we use the probability distribution function. To formulate the above problem, we apply the concept of Z-numbers and introduce a special matrix of the form diag (A , B , C) (named the generalized Z-number) where A is a fuzzy time-stamped set, B is the probability distribution function, and C is a degree of reliability of A that is described as a value of A ∗ B . Using generalized Z-numbers, we define a novel control function to investigate H–U–R stability to approximate the solution of an Apollonius-type quadratic functional equation with quality and certainty of the approximation. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to prove some new fixed-point results for the two families of multivalued dominated-contractive maps defined on a closed ball in a complete multiplicative-metric space. The fixed-point theorems for multigraph-dominated maps are established. We also investigate the existence of fixed points of the two families of graphic contractions in a multiplicative-metric space. For illustration, an example is provided to clarify that the hypothesis of the obtained result is valid. Moreover, the applications of the obtained results to show the existence of solutions to the coupled systems of nonlinear Volterra-type integral equations and mappingal equations are presented as a confirmation of the novelty of our results. [ABSTRACT FROM AUTHOR]

The objective of this paper is to introduce a new motif of α ∗ -ψ-Λ-contraction multivalued mappings, some novel fixed-point and coincidence-point results for this contraction will be investigated in the scope of F-metric spaces, and some examples are given to illustrate our main results and we derive the existence and uniqueness of a solution of a functional equation to support our main result. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove some new fixed point results for a pair of multivalued dominated locally contractive mappings in b-multiplicative metric space. Further, fixed point theorems for multigraph-dominated mappings are also established. Some new fixed point results on a closed ball are obtained for a pair of multigraph-dominated mappings endowed with graphic structure in b-multiplicative metric space. An illustrative example is given to show the validity of the hypothesis of our obtained result. Moreover, applications for a coupled system of nonlinear Volterra-type integral equations and functional equations in dynamic programming are presented to show the novelty of our results. [ABSTRACT FROM AUTHOR]

In this paper we give some hyperstability and stability results for the Cauchy and Jensen functional equations on restricted domains. We provide a simple and short proof for Brzdȩk's result concerning a hyperstability result for the Cauchy equation. [ABSTRACT FROM AUTHOR]

We investigate the Hyers–Ulam stability of the following cubic–quadratic functional equation relative to elliptic curves f (x + y + z , u + v + w) + f (x + y − z , u + v + w) + 2 f (x , u − w) + 2 f (y , v − w) = f (x + y , u + w) + f (x + y , v + w) + f (x + z , u + w) + f (x − z , u + v − w) + f (y + z , v + w) + f (y − z , u + v − w) . The function f (x , y) = x 3 + a x + b − y 2 having level curves as elliptic curves is a solution of the above functional equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new generalized p-dimensional multifarious radical reciprocal functional equation combining three classical means: arithmetic, geometric, and harmonic. Mainly, we find its general solution and stability related to the Ulam problem in modular spaces by using the fixed point method with suitable counterexamples. Importantly, in this paper, we illustrate the geometrical interpretation and applications of the introduced Pythagorean means multifarious functional equation in connection with the parallel circuit. Furthermore, we provide a formula for finding the equivalent resistance R eq of parallel electrical circuit using functional equations, which relates the electrical resistances and conductances with suitable examples. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to establish the Hyers–Ulam stability and hyperstability of a Jensen-type quadratic mapping in 2-Banach spaces. That is, we prove the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form g (x + y 2 + z) + g (x + y 2 − z) + g (x − y 2 + z) + g (x − y 2 − z) = g (x) + g (y) + 4 g (z) , in 2-Banach spaces by using the Hyers direct method. [ABSTRACT FROM AUTHOR]

In this article, we introduce some special several variables mappings which are quadratic in each variable and show that such mappings can be defined as a single equation that is the generalized multi-quadratic functional equation. We also apply a fixed point theorem to establish the Hyers–Ulam stability for the generalized multi-quadratic functional equations. Furthermore, we present an example and a few corollaries corresponding to some known stability results. [ABSTRACT FROM AUTHOR]

This research paper deals with general solution and the Hyers–Ulam stability of a new generalized n-variable mixed type of additive and quadratic functional equations in fuzzy modular spaces by using the fixed point method. [ABSTRACT FROM AUTHOR]

In this article, we study n-variable mappings which are cubic in each variable. We also show that such mappings can be described by an equation, say, multi-cubic functional equation. Furthermore, we study the stability of such functional equations in the modular space X ρ by applying Δ 2 -condition and the Fatou property (in some cases) on the modular function ρ. Finally, we show that, under some mild conditions, one of these new multi-cubic functional equations can be hyperstable. [ABSTRACT FROM AUTHOR]

It is well known that second-order information is a basic tool notably in optimality conditions and numerical algorithms. In this work, we present a generalization of optimality conditions to strongly convex functions of order γ with the help of first- and second-order approximations derived from (Optimization 40(3):229-246, 2011) and we study their characterization. Further, we give an example of such a function that arises quite naturally in nonlinear analysis and optimization. An extension of Newton's method is also given and proved to solve Euler equation with second-order approximation data. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain refined stability results and alternative stability results for additive and quadratic functional equations using direct method in modular spaces. [ABSTRACT FROM AUTHOR]

In this paper, we will obtain the optimal Hyers-Ulam's constant for the first-order linear differential equations $p(t)y'(t) - q(t)y(t) - r(t) = 0$ . [ABSTRACT FROM AUTHOR]

We introduce the notion of a modified α- ϕ-fuzzy contractive mapping and prove some results in fuzzy metric spaces for such kind of mappings. The theorems presented provide a generalization of some interesting results in the literature. Two examples and an application to integral equations are given to illustrate the usability of our theory. [ABSTRACT FROM AUTHOR]

In this paper, some new generalized retarded inequalities for discontinuous functions are discussed, which are effective in dealing with the qualitative theory of some impulsive differential equations and impulsive integral equations. Compared with some existing integral inequalities, these estimations can be used as tools in the study of differential-integral equations with impulsive conditions. [ABSTRACT FROM AUTHOR]

The functional equation related to a distance measure can be generalized as follows: where f is an information measure, P and Q are in the set of n-ary discrete complete probability, and $\sigma_{i}$ is a permutation for each $i=0, 1, \ldots, n-1$. In this paper, we investigate the superstability of the above functional equation and also four generalized functional equations: [ABSTRACT FROM AUTHOR]

In this paper, we prove a general uniqueness theorem that can easily be applied to the (generalized) Hyers-Ulam stability of a large class of functional equations, which includes monomial functional equations ( e.g. the Cauchy additive functional equation, the quadratic functional equation, and the cubic functional equation, etc.). This uniqueness theorem can save us much trouble in proving the uniqueness of relevant solutions repeatedly appearing in the stability problems for functional equations in fuzzy spaces. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the notions of $(n,\beta)$-normed space and non-Archimedean $(n,\beta)$-normed space, then we study the Hyers-Ulam stability of the Cauchy functional equation and the Jensen functional equation in non-Archimedean $(n,\beta)$-normed spaces and that of the pexiderized Cauchy functional equation in $(n,\beta)$-normed spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove a general theorem for the stability of a large class of n-dimensional functional equations of the form $Df(x_{1}, x_{2}, \ldots, x_{n}) = 0$ by applying the direct method. [ABSTRACT FROM AUTHOR]

In this paper, we prove a general uniqueness theorem that can easily be applied to the (generalized) Hyers-Ulam stability of a large class of functional equations, which includes monomial functional equations (e.g. the Cauchy additive functional equation, the quadratic functional equation, and the cubic functional equation, etc.). This uniqueness theorem can save us much trouble in proving the uniqueness of relevant solutions repeatedly appearing in the stability problems for functional equations in fuzzy spaces. [ABSTRACT FROM AUTHOR]

We first establish a new fixed point theorem for nonautonomous type superposition operators. After that, we prove the existence of integrable solutions for a general nonlinear functional integral equation in an L1 space on an unbounded interval by using our theorem. Our main tool is the measure of weak noncompactness. [ABSTRACT FROM AUTHOR]

In this paper, we extend the concept of contraction mappings in b-metric spaces and utilize this concept to prove the existence and uniqueness of fixed point theorems for such mappings in such a space. We also prove the generalized Ulam-Hyers stability and well-posed results for a fixed point equation employing the concept of α-admissibility in b-metric spaces. We shall construct some examples to support our novel results. [ABSTRACT FROM AUTHOR]

Four common fixed point theorems for a pair of weakly compatible mappings satisfying contractive conditions of integral type in metric spaces are proved. The existence result of bounded solutions for a system of functional equations arising in dynamic programming is discussed by using one of the common fixed point theorems obtained in this paper. An example is given to illustrate that our results extend properly two fixed point theorems due to Branciari and Rhoades. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the superstability of the functional equation f (pr, qs) + f (ps, qr) = θ(pq, rs)f (p, q)f (r, s) for all p, q, r, s ∈ G, where G is an Abelian group, f a functional on G2, and θ a cocycle on G². This functional equation is a generalized form of the functional equation f (pr, qs) + f (ps, qr) = f (p, q)f (r, s), which arises in the characterization of symmetrically compositive sum-form distance measures, and as products of some multiplicative functions. In reduction, they can be represented as exponential functional equations. Also we investigate the superstability with following functional equations: f (pr, qs) + f (ps, qr) = θ(pq, rs)f (p, q)g(r, s), f (pr, qs) + f (ps, qr) = θ(pq, rs)g(p, q)f (r, s), f (pr, qs) + f (ps, qr) = θ(pq, rs)g(p, q)g(r, s), f (pr, qs) + f (ps, qr) = θ(pq, rs)g(p, q)h(r, s). [ABSTRACT FROM AUTHOR]

In this paper, first, we define fuzzy C*-algebras and fuzzy Lie C*-algebras; then, using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy C*-algebras and fuzzy Lie C*-algebras for an m-variable additive functional equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new type of multivalued operators similar to those of Ciric-Suzuki type. A common fixed point theorem for multivalued maps on metric spaces satisfying Ciric-Suzuki-type inequality is proved. Applications to certain functional equations arising in dynamic programming are also discussed. [ABSTRACT FROM AUTHOR]

We prove the Hyers-Ulam stability of the Cauchy functional inequality and the Cauchy-Jensen functional inequality in 2-Banach spaces. Moreover, we prove the superstability of the Cauchy functional inequality and the Cauchy-Jensen functional inequality in 2-Banach spaces under some conditions. MSC: 39B82, 39B52, 39B62, 46B99, 46A19. [ABSTRACT FROM AUTHOR]

In this paper, the Hyers-Ulam stability of the Pexider functional equation in a non-Archimedean space is investigated, where σ is an involution in the domain of the given mapping f. MSC 2010:26E30, 39B52, 39B72, 46S10 [ABSTRACT FROM AUTHOR]

We prove the Hyers-Ulam stability of a second-order linear functional equation in single variable (with constant coefficients) that is connected with the Fibonacci numbers and Lucas sequences. In this way we complement, extend, and/or improve some recently published results on stability of that equation. [ABSTRACT FROM AUTHOR]

We prove the generalized Hyers-Ulam stability of the following additive-cubic equation f(kx + y) + f(kx - y) = kf(x + y) + kf(x - y) + 2f(kx) - 2kf(x) in the setting of random normed spaces. [ABSTRACT FROM AUTHOR]

We will investigate the superstability of the sine functional equation from the following Pexidertype functional equation f(x+y)+g(x-y) = λ∙h(x)k(y) (λ : constant),which can be considered the mixed functional equation of the sine and cosine functions, the mixed functional equation of the hyperbolic sine and hyperbolic cosine functions, and the exponential-type functional equations. [ABSTRACT FROM AUTHOR]

We obtain the superstability of the Pexiderized multiplicative functional equation f(xy) = g(x)h(y) and investigate the stability of this equation in the following form: 1/(1 + ψ(x, y)) ⩽ f(xy)/g(x)h(y) ⩽ 1 + ψ(x, y). [ABSTRACT FROM AUTHOR]

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-quartic functional equation: f(x + 2y) + f(x - 2y) + 2f(x + y) + 2f(-x - y) + 2f(x - y) + 2f(y - x) - 4f(-x) - 2f(x) + f(2y) + f(-2y) - 4f(y) - 4f(-y) in fuzzy Banach spaces. [ABSTRACT FROM AUTHOR]

We will investigate the superstability of the (hyperbolic) trigonometric functional equation from the following functional equations: f(x + y) ± g(x - y) = λf(x)g(y) and f(x + y) ± g(x - y) = λg(x)f(y), which can be considered the mixed functional equations of the sine function and cosine function, the hyperbolic sine function and hyperbolic cosine function, and the exponential functions, respectively. [ABSTRACT FROM AUTHOR]

We consider the Hyers-Ulam stability of a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions. [ABSTRACT FROM AUTHOR]

We deal with the well-posedness for solutions of nonlinear integrodifferential equations of secondorder in Hilbert spaces by converting the problem into the contraction mapping principle with more general conditions on the principal operators and the nonlinear terms and obtain a variation of constant formula of solutions of the given nonlinear equations. [ABSTRACT FROM AUTHOR]

We obtain the generalized Hyers-Ulam stability of the Cauchy-Jensen functional equation 2f(x + y, (z + w)/2) = f(x, z) + f(x,w) + f(y, z) + f(y,w). [ABSTRACT FROM AUTHOR]

We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi-β-normed spaces and then the stability by using a subadditive function for the generalized quartic function f : X → Y such that f(ax+by)+f(ax-by)-2a2(a2 -b2)f(x) = (ab)2[f(x+y)+f(x-y)]-2b2(a2 -b2)f(y), where a≠0, b≠0, a ± b≠0, for all x, y ϵ X. [ABSTRACT FROM AUTHOR]

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-quartic functional equation f(x + 2y) + f(x - 2y) = 2f(x + y) + 2f(-x - y) + 2f(x - y) + 2f(y - x) - 4f(-x) - 2f(x) + f(2y) + f(-2y) - 4f(y) - 4f(-y) in complete random normed spaces. [ABSTRACT FROM AUTHOR]

We investigate the general solution of the quadratic functional equation f(2x + y) + 3f(2x - y) = 4f(x - y) + 12f(x), in the class of all functions between quasi-β-normed spaces, and then we prove the generalized Hyers-Ulam stability of the equation by using directmethod and fixed point method. [ABSTRACT FROM AUTHOR]

Cădariu and Radu applied the fixed point theorem to prove the stability theorem of Cauchy and Jensen functional equations. In this paper, we prove the generalized Hyers-Ulam stability via the fixed point method and investigate new theorems via direct method concerning the stability of a general quadratic functional equation. [ABSTRACT FROM AUTHOR]

We investigate the superstability of the functional equation f(xy) + xf(y) + g(x)y, where f and g are the mappings on Banach algebra A. We have also proved the superstability of generalized derivations associated to the linear functional equation f(γx+βy) + γf(x)+βf(y), where γ, β ∈ C. [ABSTRACT FROM AUTHOR]

We introduce a generalized quadratic functional equation f(rx + sy) = rf(x) + sf(y) - rsf(x - y), where r, s are nonzero real numbers with r + s = 1.We show that this functional equation is quadratic if r, s are rational numbers. We also investigate its stability problem on restricted domains. These results are applied to study of an asymptotic behavior of these generalized quadratic mappings. [ABSTRACT FROM AUTHOR]

This paper deals with some inequalities for trigonometric and hyperbolic functions such as the Jordan inequality and its generalizations. In particular, lower and upper bounds for functions such as (sin x)/x and x/ sinh x are proved. [ABSTRACT FROM AUTHOR]

We obtain the generalized Hyers-Ulam stability of the bi-quadratic functional equation f(x+y, z+ w) + f(x + y, z - w) + f(x - y, z + w) + f(x - y, z - w) = 4[f(x, z) + f(x,w) + f(y, z) + (y,w)] in quasinormed spaces. [ABSTRACT FROM AUTHOR]

We prove the Hyers-Ulam stability of a 2-dimensional quadratic functional equation in a class of vector variable functions in Banach modules over a unital C*-algebra. [ABSTRACT FROM AUTHOR]